Dividing a rectangle into squares

This is a rather simple question intuitively, but I can't seem to find a rigorous explanation for this: suppose we are diving a rectangle into some number of equal sized squares, then they must all be aligned together(such that we can produce theses squares by performing horizontal and vertical cuts on the rectangle).

It makes sense to me, since if we have misalignment, then eventually we are going to reach a position where we can't fit the square. However, this assumes that the squares are originally aligned in the first place... Any rigorous explanation/counter example to this?

• You should always be able to tesselate a rectangle with equally-sized squares as long as the ratio of length to width is a rational number. A counter example would be a rectangle with length $\sqrt{2}$ and width $1$. – D.B. May 25 '18 at 19:11
• To be clear -- you're asking if there's a way to tile a rectangle with squares other than a regular pattern? – dbx May 25 '18 at 19:19
• @dbx yes, I was wondering if there's a way to tile a rectangle with equal sized squares other than the regular grid pattern, but it seems that there will be no such ways since the answers suggest to orient the squares in the corners and then the right angles will force the squares to be in a grid. – Jesse Meng May 25 '18 at 19:47